Average Error: 3.6 → 0.1
Time: 11.3s
Precision: 64
\[-14 \le a \le -13 \land -3 \le b \le -2 \land 3 \le c \le 3.5 \land 12.5 \le d \le 13.5\]
\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
↓
\[2 \cdot \mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\left(b + c\right) + \left(d + a\right)\right)\right)\right)\right)\]
\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2
↓
2 \cdot \mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\left(b + c\right) + \left(d + a\right)\right)\right)\right)\right)
double f(double a, double b, double c, double d) {
        double r2359469 = a;
        double r2359470 = b;
        double r2359471 = c;
        double r2359472 = d;
        double r2359473 = r2359471 + r2359472;
        double r2359474 = r2359470 + r2359473;
        double r2359475 = r2359469 + r2359474;
        double r2359476 = 2.0;
        double r2359477 = r2359475 * r2359476;
        return r2359477;
}
↓
double f(double a, double b, double c, double d) {
        double r2359478 = 2.0;
        double r2359479 = b;
        double r2359480 = c;
        double r2359481 = r2359479 + r2359480;
        double r2359482 = d;
        double r2359483 = a;
        double r2359484 = r2359482 + r2359483;
        double r2359485 = r2359481 + r2359484;
        double r2359486 = expm1(r2359485);
        double r2359487 = log1p(r2359486);
        double r2359488 = r2359478 * r2359487;
        return r2359488;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original3.6
Target3.8
Herbie0.1
\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2\]

Derivation

  1. Initial program 3.6

    \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
  2. Using strategy rm

  3. Applied associate-+r+2.8

    \[\leadsto \left(a + \color{blue}{\left(\left(b + c\right) + d\right)}\right) \cdot 2\]
  4. Using strategy rm

  5. Applied log1p-expm1-u2.8

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(a + \left(\left(b + c\right) + d\right)\right)\right)\right)\right)} \cdot 2\]
  6. Using strategy rm

  7. Applied add-cbrt-cube2.9

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\sqrt[3]{\left(\mathsf{expm1}\left(\left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \mathsf{expm1}\left(\left(a + \left(\left(b + c\right) + d\right)\right)\right)\right) \cdot \mathsf{expm1}\left(\left(a + \left(\left(b + c\right) + d\right)\right)\right)}\right)}\right) \cdot 2\]

  8. Taylor expanded around -inf 3.7

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(e^{a + \left(b + \left(c + d\right)\right)} - 1\right)}\right) \cdot 2\]

  9. Simplified0.1

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\mathsf{expm1}\left(\left(\left(a + d\right) + \left(b + c\right)\right)\right)\right)}\right) \cdot 2\]

  10. Final simplification0.1

    \[\leadsto 2 \cdot \mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\left(b + c\right) + \left(d + a\right)\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (a b c d)
  :name "Expression, p6"
  :pre (and (<= -14 a -13) (<= -3 b -2) (<= 3 c 3.5) (<= 12.5 d 13.5))

  :herbie-target
  (+ (* (+ a b) 2) (* (+ c d) 2))

  (* (+ a (+ b (+ c d))) 2))